How do the Navier$-$Stokes equations serve as a foundational theory in chemical engineering? Discuss their significance in understanding fluid behavior, particularly in viscous flows encountered in chemical processes. Analyze the challenges posed in solving these equations, such as defining accurate boundary conditions and modeling turbulent flows. Furthermore, explore their practical relevance in engineering applications within the chemical industry, including reactor design, mixing processes, and fluid transport systems. Illustrate how a thorough understanding of the Navier$-$Stokes equations enhances efficiency, safety, and product quality in chemical engineering operations.